This is something I learned and geometry class and completely forgot for a while. All you will need is a drafting/drawing compass or a large homemade one for big circles. I have a little drafting kit that has came in very useful in engineering school! The steps are very short and will be best shown if you view all steps at once. Sorry if the pictures are blurry I don't have very good light to take pictures at my desk. Lets get started!
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I'm going to use a piece of paper, since it will easily show the pencil marks, and draw a circle.
First, set your compass to just over the radius of the circle.
Second, place the pin end of the compass on the and draw an arc in the middle of the circle. I have made a big black dot where I placed the compass pin and will continue to do this.
Now on the opposite side of the circle where you placed the compass, create another arc keeping the compass spread larger than the circle's radius. You don't have to be exactly across from where you started as you can be fairly inaccurate to get good results.
Now we are going to make an arc perpendicular to the two we just made.
Now create another arc on the top of the circle directly across from the one made on the bottom.
Where two arcs intersect towards their ends you will need to draw a line between the two intersections.
Finally, connect the intersections of the remaining arcs. Where lines 5 and 6 intersect each other is the center of your circle.
If you are needing to drill a hole in the new found center make sure to either start with a small drill bit or use a punch to keep the drill bit from "walking" when initially drilling the hole. If you are drilling through metal using a punch is necessary even when starting with a small drill bit. Lastly, no one uses drill bits properly but few properly maintain their bits by keeping them sharp. Buying a Drill Doctor will save you loads of time and money by keeping your bits nice and sharp. I tried hand sharpening drill bits but couldn't get it despite my machinist friend's help, so I spent the money on a great product. Best $80 I have spent thus far!
Quick note on variability with this method. Intersecting arcs need to be drawn with the same radius or compass length, ex: 1&2 or 3&4 but not all four. Tighten up your compass to it hold the length between placements if this is a problem. You don't need to make the second set of arcs perpendicular to the first. When making one set of intersecting arcs you don't have to place the compass in the exact location 180 degrees away but try to be close. I have attached a perspective free scan showing what you could get away with.
If you have any questions feel free to ask! If there are problems please include enough information to clearly outline what step/part, what I did and how it is an issue.